Theorem eqglact 17468

Description: A left coset can be expressed as the image of a left action. (Contributed by Mario Carneiro, 20-Sep-2015.)

Hypotheses

Ref	Expression
eqger.x	⊢ 𝑋 = (Base‘𝐺)
eqger.r	⊢ ∼ = (𝐺 ~QG 𝑌)
eqglact.3	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
eqglact	⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌))

Distinct variable groups:   𝑥, +   𝑥, ∼   𝑥,𝐺   𝑥,𝑋   𝑥,𝐴   𝑥,𝑌

Proof of Theorem eqglact

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqger.x	. . . . . . 7 ⊢ 𝑋 = (Base‘𝐺)
2		eqid 2610	. . . . . . 7 ⊢ (invg‘𝐺) = (invg‘𝐺)
3		eqglact.3	. . . . . . 7 ⊢ + = (+g‘𝐺)
4		eqger.r	. . . . . . 7 ⊢ ∼ = (𝐺 ~QG 𝑌)
5	1, 2, 3, 4	eqgval 17466	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝐴 ∼ 𝑥 ↔ (𝐴 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
6		3anass 1035	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌) ↔ (𝐴 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
7	5, 6	syl6bb 275	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) → (𝐴 ∼ 𝑥 ↔ (𝐴 ∈ 𝑋 ∧ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌))))
8	7	baibd 946	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴 ∼ 𝑥 ↔ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
9	8	3impa 1251	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴 ∼ 𝑥 ↔ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
10	9	abbidv 2728	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → {𝑥 ∣ 𝐴 ∼ 𝑥} = {𝑥 ∣ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)})
11		dfec2 7632	. . 3 ⊢ (𝐴 ∈ 𝑋 → [𝐴] ∼ = {𝑥 ∣ 𝐴 ∼ 𝑥})
12	11	3ad2ant3 1077	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = {𝑥 ∣ 𝐴 ∼ 𝑥})
13		eqid 2610	. . . . . . . . 9 ⊢ (𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥))) = (𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))
14	13, 1, 3, 2	grplactcnv 17341	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴):𝑋–1-1-onto→𝑋 ∧ ◡((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘((invg‘𝐺)‘𝐴))))
15	14	simprd 478	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ◡((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘((invg‘𝐺)‘𝐴)))
16	13, 1	grplactfval 17339	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑋 → ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)))
17	16	adantl 481	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)))
18	17	cnveqd 5220	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ◡((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)))
19	1, 2	grpinvcl 17290	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
20	13, 1	grplactfval 17339	. . . . . . . 8 ⊢ (((invg‘𝐺)‘𝐴) ∈ 𝑋 → ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘((invg‘𝐺)‘𝐴)) = (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)))
21	19, 20	syl 17	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝑔 ∈ 𝑋 ↦ (𝑥 ∈ 𝑋 ↦ (𝑔 + 𝑥)))‘((invg‘𝐺)‘𝐴)) = (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)))
22	15, 18, 21	3eqtr3d 2652	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) = (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)))
23	22	cnveqd 5220	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ◡◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) = ◡(𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)))
24	23	3adant2 1073	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → ◡◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) = ◡(𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)))
25	24	imaeq1d 5384	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → (◡◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌) = (◡(𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)) “ 𝑌))
26		imacnvcnv 5517	. . 3 ⊢ (◡◡(𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌) = ((𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌)
27		eqid 2610	. . . . 5 ⊢ (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)) = (𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥))
28	27	mptpreima 5545	. . . 4 ⊢ (◡(𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)) “ 𝑌) = {𝑥 ∈ 𝑋 ∣ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌}
29		df-rab 2905	. . . 4 ⊢ {𝑥 ∈ 𝑋 ∣ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌} = {𝑥 ∣ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)}
30	28, 29	eqtri 2632	. . 3 ⊢ (◡(𝑥 ∈ 𝑋 ↦ (((invg‘𝐺)‘𝐴) + 𝑥)) “ 𝑌) = {𝑥 ∣ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)}
31	25, 26, 30	3eqtr3g 2667	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌) = {𝑥 ∣ (𝑥 ∈ 𝑋 ∧ (((invg‘𝐺)‘𝐴) + 𝑥) ∈ 𝑌)})
32	10, 12, 31	3eqtr4d 2654	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → [𝐴] ∼ = ((𝑥 ∈ 𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596  {crab 2900   ⊆ wss 3540   class class class wbr 4583   ↦ cmpt 4643  ^◡ccnv 5037   “ cima 5041  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  [cec 7627  Basecbs 15695  +_gcplusg 15768  Grpcgrp 17245  inv_gcminusg 17246   ~_QG cqg 17413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-ec 7631  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-eqg 17416

This theorem is referenced by:  eqgen  17470  cldsubg  21724  tgpconcompeqg  21725  snclseqg  21729